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I know that you can use $\pm$ for when the answer could be either positive or negative, e.g., $x^2=16$, $x=\pm 4$.

But is there a symbol that implies that you use both the positive and the negative values? For example, I want to do something along the lines of:

$$(2/3a) \left(\sqrt[3]{2b^3 - 9abc + \sqrt{−4(b^2−3a)}} + \sqrt[3]{2b^3 - 9abc - \sqrt{−4(b^2−3a)}}\right)$$

It would be very useful to not have to write out the cube root twice and instead have a plus and minus sign before the square root.

J W
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Alex Jones
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    I'm not sure what you mean. Do you want, for example, a way to write$$(\sqrt2-\sqrt3-\sqrt5) \times(\sqrt2-\sqrt3+\sqrt5)\times\ (\sqrt2+\sqrt3-\sqrt5) \times(\sqrt2+\sqrt3+\sqrt5)$$concisely? – Akiva Weinberger Jan 07 '16 at 15:33
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    You can use $(-1)^n$ in order to change the sign of "everything". – JnxF Jan 07 '16 at 15:35
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    @Alex: You could try, $$\sum_{n=0}^1 2\big(a+(-1)^nb\big)$$ – Tito Piezas III Jan 07 '16 at 15:37
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    I 'm not sure your example exactly explain what you need, if you have a long expression why not writing $ 2 \times (a+b) + 2 \times (a- b) =2 \times (2 a)$ ? – Nizar Jan 07 '16 at 15:39
  • (PS. That big expression up there is actually an integer, $-24$.) – Akiva Weinberger Jan 07 '16 at 15:39
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    I don't know if it helps but my actual expression is along the lines of (2/3a) * (cube root( 2b^3 - 9abc + $\sqrt{ -4(b^2-3a)}$ ) + cube root( 2b^3 - 9abc - $\sqrt{ -4(b^2-3a)}$ )) and I would prefer not to write out the whole cube root each time and just write it once with a plus and minus sign before the square root sign. (sorry about the formatting) – Alex Jones Jan 07 '16 at 15:51
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    Although all answers work, I plead people to NOT abbreviate two- or three- term sums, especially with summation symbols. Conciseness at the expense of readability is not appreciated by any reader. Rather, give names to the messy expressions under the root symbols and leave the sum as it is. Of all the answers, the only one that would not make me foam at the mouth is @celtschk 's. – guest Jan 07 '16 at 22:06
  • What @guest said. – TonyK Jan 08 '16 at 16:55

4 Answers4

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Another way to write it is:

$$\frac{2}{3a}\left(R_+ + R_-\right) \text{ where } R_\pm = \sqrt[3]{2b^3-9abc\pm\sqrt{-4(b^2-3a)}}$$

celtschk
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There's no single symbol for it. So just use, $$\sum_{n=0}^1\sqrt[3]{2b^3-9abc+(-1)^n\sqrt{-4(b^2-3a)}}$$

Tito Piezas III
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No, and there's a good reason for it: it cannot convey the necessary information.
How would the reader know that the intention is to add?
What if you intended the "and" to be for multiplication?
You need to denote the operator somehow, and that will take care of the "and" part by itself.

user541686
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You can do this with multisets:

$$\sum(2×(a \pm b))$$

is the expression you want.

Basically, $\pm b$ is the multiset $\{b,-b\}$, so $a \pm b$ is $\{a+b,a-b\}$, so $2 \times (a \pm b)$ is the multiset $\{2(a+b),2(a-b)\}$, then the $\sum$ just means "take the sum."

Edit. Let me add that this notation, being not entirely standard, should never be used without prior explanation. Putting all that side, however, I want to draw attention to a cultural problem here. That problem, in short, is a general aversion to new and interesting ways of denoting our thoughts and ideas. This aversion is holding mathematics back; see here, for example. Here's how it plays out in practice.

What should happen.

The reader encounters a new and unfamiliar notation. Suddenly intrigued by the possibility of denoting his ideas more tersely and clearly, he does his best to understand the conventions of that notation and/or any deeper ideas on which it is predicated, and spends some time experimenting with it. He writes a few proofs in the new and unfamiliar language. After the new language has become sufficiently familiar, he makes a judgement regarding whether or not the benefits of the new notation outweigh the costs, and makes a conscious decision to either adopt it in his own work and writing, or not to adopt it. If he chooses the former, he is unconcerned about the potential reduction in readership, because he knows that truly elite mathematicians are intellectually flexible, and that in fact, most people are quite flexible once they've adopted the right mindset. He therefore knows that, by adopting the best possible conventions that he can, he is fundamentally doing other people a favor, and that and while this may infuriate some, nonetheless the benefits of adopting the best possible conventions outweigh the costs, and that is that.

What tends to happen.

The reader encounters a new and unfamiliar notation. The parts of his brain that are responsible for tribal thinking instantly categorize the person using the notation as an "outsider" whose opinions and ideas fundamentally don't matter (unless they're already high-up in the mathematical community, in which its an automatic movement to "what should happen"). He thinks to himself: how dare she write this kind of drivel? He starts to feel angry, and he is convinced that his anger is rational and justified. Already, whether or not the new notation could be useful to him - or to mathematics - has ceased to matter. Non-standard notation! he growls. The tools of cowards! The fact that the writer, in all likelihood, is using this notation precisely because she found it to be useful has ceased to matter. All that matters is how best to attack this new and unfamiliar experience. After composing himself for a moment, he decides on the lecturing approach. He will simply talk down to the other person, until she finally "gets" that pandering to people's inflexibility is a good and noble pursuit. He leaves a comment to the effect that: "Look, if you don't want to reach the largest possible audience, then keep on writing that way." The OP reads this comment with great sadness. They could have learned something, she thinks to herself. Instead, they have learned nothing.

Take home message.

Attitudes that justify notational or intellectual stagnation deserve to be regarded with both suspicion and sadness (in perhaps equal parts), especially when they're predicated on tribal thinking in which the ingroup is privileged and ideas from the outgroup ignored. We should be careful to reject those attitudes that, by retarding progress, make mathematics worse, while consciously adopting beliefs that make it better.

Community
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goblin GONE
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    I personally would not understand what this notation is meant to signify (I am a professional mathematician), and I would be concerned that most readers wouldn't understand it either. I've never before seen the interpretation that $\pm$ defines a multiset. – Nate Eldredge Jan 07 '16 at 19:39
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    Downvoting because I would never expect this to be a notation about multisets, and would complain if this notation was used without an explanation that I was supposed to interpret this in this way. – Stella Biderman Jan 07 '16 at 21:43
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    When $\sigma$ is used to denote a sum, the MathJax code is \sum. It makes it bigger, and it can have limits in display mode. – Akiva Weinberger Jan 07 '16 at 21:45
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    Another problem is that, in set-like contexts, $\times$ usually means Cartesian product, not scalar multiplication. – David Richerby Jan 08 '16 at 00:27
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    +1. @NateEldredge: I would argue that your aversion to this interpretation is a bit unfounded; the typical "or" interpretation is just a colloquial version of the multiset interpetation. It's just that simply no one called it a multiset before, but that's what it is. How else would you formally define it? – user541686 Jan 08 '16 at 19:39
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    @Mehrdad: I'm not saying there's anything inherently wrong with that interpretation, and anyway, whether or not I personally have an "aversion" to it is beside the point. I am saying that, in my professional opinion, the majority of mathematical readers are not likely to understand it to mean what's intended; at least, not without very careful explanation. In my experience, mathematicians don't attach any formal meaning to the $\pm$ symbol; it's used only as informal shorthand. – Nate Eldredge Jan 09 '16 at 00:16
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    In other words, I think this notation is trying to be much too clever. Whether or not it "technically" means the right thing is irrelevant if readers don't manage to figure out that it does. The language of mathematics not only has syntax and semantics but also idiom. Regardless of whether it has the right literal meaning, I would argue this notation is certainly not idiomatic, and on that basis alone, I would avoid it. – Nate Eldredge Jan 09 '16 at 00:20
  • The part of this post after "What should happen" is entirely irrelevant to the question that was asked here. It should be relegated to comments or omitted entirely. – Milo Brandt Jan 09 '16 at 02:19
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    @MiloBrandt: I would suggest it could go to chat. It's a reasonable point for discussion, but of course the main site is only for Q&A and not discussion. – Nate Eldredge Jan 09 '16 at 02:40
  • Every downvote I gain from here on is evidence that I am right. There really is a serious cultural problem here, and shame on those who wont admit it. – goblin GONE Jan 09 '16 at 05:19
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    «Every downvote I gain from here on is evidence that I am right. There really is a serious cultural problem here, and shame on those who wont admit it» is the sort of thing that raises one's crackpot index. – Mariano Suárez-Álvarez Jan 09 '16 at 05:23
  • @Mariono, no, it doesn't, because I'm not advocating a particular technical idea, and I'm certainly not saying anything is revolutionary, and I'm DEFINITELY not claiming to have actually made any revolutionary contributions. Rather, I'm pointing to a system of attitudes that demonstrably exist, that are obviously problematic, and I'm saying that every downvote here is evidence that these attitudes are out there. Therefore, I think that whoever upvotes your comment, their brain has too many connections and insufficiently many distinctions, which by the way is the hallmark of crankery. – goblin GONE Jan 09 '16 at 05:34
  • @Mariano, in short, then, I don't think you quite know what you're talking about. – goblin GONE Jan 09 '16 at 05:41
  • @NateEldredge: A couple things: (1) Could you explain what interpretation the professional mathematician does assign to the plus/minus symbol? Surely they've seen it before, right? How do they interpret it when they come across it? Or is their only standard response to burn the paper in flames without any attempt to understand it and then move on? (2) Could you also explain why you think the question and answers on this page should be directed toward the "professional mathematician" in the first place? Are you sure you're not confusing this website with MathOverflow? – user541686 Jan 09 '16 at 05:42
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    @MarianoSuárez-Alvarez: I agree that wasn't the best way to phrase it, but goblin does have a point. It's a genuine problem. I have empirically seen similar reactions myself; for whatever reason, some people here do have a habit of knee-jerk downvoting answers merely because of unfamiliarity, and it's awful. I'd even go so far as to say that right now I can almost guarantee you that some of the downvotes this answer is receiving at this point are because people don't like the comments, and have nothing to do with the merits of the answer itself. – user541686 Jan 09 '16 at 05:50
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    @Mehrdad 79% of the text is irrelevant to the question, it refers to "cultural problems" that "hold mathematics back", it talks about the " tribal" part of the brain in order to characyerize an imagined "oponent", clearly the oppressor, it has this ”defending the outsider» theme... and on top of all that, it advocates a notation which is rather bad. I have very little problem understanding the downvotes. – Mariano Suárez-Álvarez Jan 09 '16 at 05:55
  • @Mariano, if you are claiming that what I've written about ingroup bias is incorrect, you've probably never studied scientific psychology. If you're saying that this problem should not be brought to light and discussed, then you're my enemy - and an enemy of progress, too. – goblin GONE Jan 09 '16 at 05:59
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    Heh. OK. I'll add that to my CV. An enemy of progress! Do I get a superpower with that? – Mariano Suárez-Álvarez Jan 09 '16 at 06:01
  • @MarianoSuárez-Alvarez: Yes I agree that I don't have any problem understanding the downvotes that came after the modification to the answer, but those are only the ~3 or so since 4 hours ago. I was talking about the ~9 or so other downvotes (the answer had zero net votes when I wrote my comment, and while I didn't check the number then, I don't expect it has received many/any upvotes since then); I do have a problem understanding those and I think they were unfounded knee-jerk reactions like I talked about earlier. – user541686 Jan 09 '16 at 06:10
  • I have made a chat room: http://chat.stackexchange.com/rooms/34059/discussion-of-pm I suggest that all further discussion take place there. – Nate Eldredge Jan 09 '16 at 08:13